Vladimír Lisý

A note on 'Langevin theory of anomalous Brownian motion made simple'

In our recent paper (Tothova et al 2011 Eur. J. Phys. 32 645), we extensively used a rule allowing us to convert linear stochastic equations of motion for the position of a Brownian particle to deterministic equations for its mean square displacement. This rule was established in a little known and hardly accessible work (Vladimirsky 1942 Z. Eksp. Teor. Phys. 12 199, in Russian), and so far it has not been used in solving the generalized Langevin equations with memory. To make our paper more self-contained and readable for students, we present a very simple substantiation of our approach, which is suitable for the description of both normal and anomalous Brownian motion.

BROWNIAN OSCILLATORS DRIVEN BY CORRELATED NOISE IN A MOVING TRAP

Brownian Oscillators Driven by Correlated Noise in a Moving Trap Brownian oscillator, ie a micron-sized or smaller particle trapped in a thermally fluctuating environment is studied. The confining harmonic potential can move with a constant velocity. As distinct from the standard Langevin theory, the chaotic force driving the particle is correlated in time. The dynamics of the particle is described by the generalized Langevin equation with the inertial term, a coloured noise force, and a memory integral. We consider two kinds of the memory in the system. The first one corresponds to the exponentially correlated noise and in the second case the memory naturally arises within the Navier-Stokes hydrodynamics. Exact analytical solutions are obtained in both the cases using a simple and effective method not applied so far in this kind of problems.

Attenuation of the NMR signal due to hydrodynamic Brownian motion

Abstract Nuclear magnetic resonance (NMR) is a widely used nondestructive method to study random motion of spin-bearing particles in different systems. In the long-time limit the theoretical description of the NMR experiments is well developed and allows proper interpretation of measurements of normal and anomalous diffusion. The traditional description becomes, however, insufficient for the shorter-time dynamics of the particles. In the present paper, the all-time attenuation function of the NMR signal in a magnetic-field gradient due to the Brownian motion (BM) of particles in incompressible liquids is calculated by using the method of accumulation of phases by a precessing magnetic moment, without reference to a concrete model of the stochastic dynamics. The obtained expressions are then used to evaluate the attenuation within the hydrodynamic theory of the BM. It is shown that the well-known time behavior of the formulas corresponding to the Einstein theory of diffusion in the case of steady gradient and Hahns echo experiments is reached at times much larger than the characteristic time of the loss of memory in the particle dynamics. At shorter times the attenuation function significantly differs from the classical formulas used to interpret these experiments.

Attenuation of the NMR signal in a field gradient due to stochastic dynamics with memory

The attenuation function S(t) for an ensemble of spins in a magnetic-field gradient is calculated by accumulation of the phase shifts in the rotating frame resulting from the displacements of spin-bearing particles. The found S(t), expressed through the particle mean square displacement, is applicable for any kind of stationary stochastic motion of spins, including their non-markovian dynamics with memory. The known expressions valid for normal and anomalous diffusion are obtained as special cases in the long time approximation. The method is also applicable to the NMR pulse sequences based on the refocusing principle. This is demonstrated by describing the Hahn spin echo experiment. The attenuation of the NMR signal is also evaluated providing that the random motion of particle is modeled by the generalized Langevin equation with the memory kernel exponentially decaying in time. The models considered in our paper assume massive particles driven by much smaller particles.

Comment on “Motional Averaging of Nuclear Resonance in a Field Gradient”

In the Letter by Nanette N. Jarenwattananon and Louis-S. Bouchard [PRL 114, 197601 (2015)] an NMR experiment on gases in the presence of a magnetic-field gradient has been considered. As distinct from the traditional description of molecular self-diffusion, the authors calculate the decoherence of the signal taking into account the histories of molecular displacements. For this purpose the generalized Langevin equation (GLE) is applied. We show that the use of this equation is inappropriate. The calculations performed in the Letter are not correct and do not lead to the reported revised expression for line broadening that takes into account the autocorrelation effects in the diffusion process. The surprising temperature behavior of the observed NMR signal is thus not explained. In particular, the linewidth does not follow the power law f ~ T^(-1/2) at high temperatures. We give also Remarks on the Jarenwattananon and Bouchard Reply [Phys. Rev. Lett. 117, 249702 (2016)].

THERMAL FLUCTUATIONS IN ELECTRIC CIRCUITS AND THE BROWNIAN MOTION

Thermal Fluctuations in Electric Circuits and the Brownian Motion In this work we explore the mathematical correspondence between the Langevin equation that describes the motion of a Brownian particle (BP) and the equations for the time evolution of the charge in electric circuits, which are in contact with the thermal bath. The mean quadrate of the fluctuating electric charge in simple circuits and the mean square displacement of the optically trapped BP are governed by the same equations. We solve these equations using an efficient approach that allows us converting the stochastic equations to ordinary differential equations. From the obtained solutions the autocorrelation function of the current and the spectral density of the current fluctuations are found. As distinct from previous works, the inertial and memory effects are taken into account.

NMR signals within the generalized Langevin model for fractional Brownian motion

Abstract The methods of Nuclear Magnetic Resonance belong to the best developed and often used tools for studying random motion of particles in different systems, including soft biological tissues. In the long-time limit the current mathematical description of the experiments allows proper interpretation of measurements of normal and anomalous diffusion. The shorter-time dynamics is however correctly considered only in a few works that do not go beyond the standard memoryless Langevin description of the Brownian motion (BM). In the present work, the attenuation function S (t) for an ensemble of spin-bearing particles in a magnetic-field gradient, expressed in a form applicable for any kind of stationary stochastic dynamics of spins with or without a memory, is calculated in the frame of the model of fractional BM. The solution of the model for particles trapped in a harmonic potential is obtained in an exceedingly simple way and used for the calculation of S (t). In the limit of free particles coupled to a fractal heat bath, the results compare favorably with experiments acquired in human neuronal tissues. The effect of the trap is demonstrated by introducing a simple model for the generalized diffusion coefficient of the particle.

Viscosity Measurements of Dilute Poly(2-ethyl-2-oxazoline) Aqueous Solutions Near Theta Temperature Analyzed within the Joint Rouse-Zimm Model

The steady-state shear viscosity of low-concentrated Poly(2-ethyl-2-oxazoline) (PEOX) aqueous solutions is measured near the presumed theta temperature using the falling ball viscometry technique. The experimental data are analyzed within the model that joins the Rouse and Zimm bead-spring theories of the polymer dynamics at the theta condition, which means that the polymer coils are considered to be partially permeable to the solvent. The polymer characteristics thus depend on the draining parameter h that is related to the strength of the hydrodynamic interaction between the polymer segments. The Huggins coefficient was found to be 0.418 at the temperature 20°C, as predicted by the theory. This value corresponds to h = 2.92, contrary to the usual assumption of the infinite h. This result indicates that the theta temperature for the PEOX water solutions is 20°C rather than 25°C in the previous studies. The experimental intrinsic viscosity is well described coming from the Arrhenius equation for the shear viscosity.

Comment on "Rheological Properties of Polyethylene Glycol (PEG 35000): An Interpretation of a Negative Intrinsic Viscosity and a High Huggins Coefficient Value." J. Macromol. Sci., Part B: Physics 2014 (53) 391"

It is shown that the recently reported negative intrinsic viscosity and the high Huggins coefficient value of polyethylene glycol (PEG 35000) water solutions [J. Macromol. Sci., Part B: Physics 2014, 53 (3) 391] is just the result of omitting some experimental data from the consideration. Viscometric studies, which we have performed on the PEG 35000 solutions, exclude an anomalous behavior of the viscosity and related quantities under the conditions of the discussed experiments.

Influence of external factors on optical parameters in Cu6PS5I thin films

Cu6PS5I superionic thin films were deposited onto silicate glass substrates by non-reactive radio frequency magnetron sputtering. Optical transmission spectra of annealed and implanted by sulphur and phosphorous thin films were investigated. In the temperature interval 77–300 K the temperature behavior of optical absorption spectra and dispersion of refractive index of annealed films as well as the influence of ionic implantation on spectra and parameters of exponential absorption edge were studied. Temperature dependences of optical pseudogap, Urbach energy and refractive index of annealed and implanted Cu6PS5I thin films were analysed.